Rate of convergence, no power series I have a question with respect to the radius of convergence of series that are not power series in them can talk about radius of convergence?, By example
$ \sum_{n=1}^{\infty } (\ln(x))^n  $
I know that this series converges when $ | \ ln (x) | <1 $ y this is $ -1 <ln (x) <$ 1 to $ e^{-1} <x <e $ 
therefore have the interval of convergence but the radius of convergence exists and if so what is it? Can talk radio series convergence power series are not?
Excuse my English, I'm confused with that.
 A: A power series automatically comes with a center for its circle of convergence.
For any other series, you could ask the question "What is the maximal radius of convergence around the point $a$?" which is a meaningful question. In particular for your example, you could put $a=1$ and get $1-\frac 1e$ as answer.
However, you do not seem to appreciate that a radius of convergence of a power series is assigned to an actual circle in the complex plane, it really is bad practice to use this name for a real convergence interval for a series which is not a power series because in that case this does not generalize to a circle.
Another issue is the fact that other types of series come with other natural shapes of convergence regions. For example, Dirichlet series are best investigated by searching for their half-plane of convergence instead of a circle of convergence.
So, in summary, you could talk about a radius of convergence, but it is not the right concept.
Also, a rate of convergence is something else, entirely.
